In recent years the development of laser-induced infrared photothermal radiometry (PTR) of semiconductors in our laboratory [1–9] and elsewhere [10] as a quantitative methodology for the measurement of transport properties of semiconductors has led to several advances in the non-contact measurement of four transport parameters: bulk recombination lifetime, (two) surface recombination velocities and carrier diffusion coefficient in Si [1–10] and GaAs [11]. Reviews of the subject matter have been presented by Mandelis [12] and Christofides et al. [13]. The major advantage of PTR over other photothermal techniques, such as photomodulated thermoreflectance (PMOR), has been found to be the higher sensitivity of PTR to the photo-excited free carrier-density-wave (the modulated-laser driven oscillating electronic diffusion wave [14]) than PMOR [15,16]. This advantage exists due to domination of the free-carrier wave over the superposed thermal-wave (TW) contributions to the PTR signal. Even so, the ever-present thermal-wave contributions due to direct lattice absorption, followed by non-radiative energy conversion and black-body (thermal infrared) emissions, have resulted in PTR signal interpretational and computational difficulties due to the large number of variables involved [5].
Therefore, confidence in the measured values of the four electronic transport properties is always accompanied by the hurdle of assuring uniqueness of the measured set of parameters in any given situation. With our development of the PTR methodology as a quantitative technique for non-destructive semiconductor diagnostics, we found [4,5] that early measurements reported without regard to computational uniqueness [17] using simplified theoretical fits to frequency-scan signals cannot be unique and therefore reliable.
Several schemes to enhance the photo-excited free carrier-density-wave (or simply “carrier-wave”, CW) contributions to the photothermal signal have been proposed, such as working in the high-frequency, CW-dominated, regime with PTR [12], or using a tightly focused pump laser beam in PMOR [18]. However, the presence of even a diminished TW component in high-frequency PTR has been shown [19] to have significant effects on the measured values of the transport parameters, to compromise sensitivity to the carrier wave and to complicate the task of physical interpretation of the signal, thus raising the question of uniqueness of the measured set of solid-state transport parameters.
On the other hand, very tight focusing of the pump laser beam in PMOR tends to give rise to usually undesirable non-linear thermal and electronic effects [15,20,21], besides being unable to sufficiently eliminate the TW component of the signal [22]. Therefore, given the fundamental and practical importance of developing an all-optical, non-destructive and non-intrusive diagnostic methodology for monitoring only the transport properties of semiconductors, we concluded that the search for a purely carrier-wave laser-based detection methodology must move in the direction of isolating and eliminating the superposition of thermal-wave contributions to the infrared emission spectrum. In view of the inability of photothermal semiconductor diagnostic methods [13,18] to eliminate the thermal-wave contributions, the development of infrared laser radiometry of semiconductors to optimize this task has been very promising, given the intrinsically higher sensitivity of its photothermal embodiment, PTR, to the photo-excited carrier density-wave than other photothermal techniques, notably PMOR [16].
In a photo-excited semiconductor of bandgap energy EG, an externally incident optical source such as a laser beam with super-bandgap energy photons hvvis/EG will be absorbed and can generate free carriers which may subsequently follow several deexcitation pathways as shown in FIG. 1 for an n-type material. Ultrafast decay to the respective bandedge (e.g. conduction band) through nonradiative transitions and emission of phonons, will raise the temperature of the semiconductor locally. The free carriers will further diffuse within their statistical lifetime and will recombine with carriers of the opposite sign across the bandgap or into impurity and/or defect states within the bandgap. The electron-hole recombination mechanism with or without phonon assistance will lead either to nonradiative energy conversion through phonon emissions (e.g. in indirect-gap semiconductors such as Si) which will further raise the temperature, or to radiative decay which will produce photons of near- or sub-bandgap energy. A table of radiative recombination lifetimes at 300 K in Si and other semiconductors has been compiled by Hall [23]. In the presence of impurity or defect states within the bandgap, free-carrier decay to one or more of those states may also occur through nonradiative or radiative transitions symbolized by dashed and full arrows, respectively, in FIG. 1. Again, the former will raise the temperature of the semiconductor crystal through phonon coupling to the lattice, whereas the latter will produce photons of energy EG−ED≅hvIR. In actual semiconductor materials, there may be a distribution of impurity and defect states into which de-excitation may occur.
Therefore, it is more relevant to consider the full spectral range of IR emissions from a photo-excited semiconductor crystal: hvIR=hv(λD). If the exciting super-bandgap radiation is intensity-modulated at frequency f=ω/2π, then the photo-generated free carrier density constitutes a spatially damped carrier-density wave (CW) (or carrier-diffusion wave [14]), which oscillates diffusively away from the generating source under its concentration gradient and recombines with a phase lag dependency on a delay time equal to its statistical lifetime, τ, a structure- and process-sensitive property [24]. FIG. 2 shows a virtual cross-section of a semiconductor Si wafer where an infrared emission photon distribution is produced following laser radiation absorption and carrier-wave generation. For one-dimensional geometries, such as those obtained with thin crystals and/or use of laser beams of large spotsize, only forward- and backward-emitted photons of wavelength λ are depicted. The IR power generated at λ within a spectral bandwidth dλ is given bydPj(z,t;λ)={WNR[TT(z,t);λ]+ηRWeR(λ)}jdλ;j=r,t[W]  (1)where WNR[TT(z,t);λ] is the thermal infrared power per unit wavelength generated due to temperature rise following optical absorption, as well as due to other nonradiative decays. The subscripts (r,t) indicate back-propagating (“reflected”) or forward-propagating (“transmitted”) photon power. WeR(λ) is the spectral power per unit wavelength, the product of the recombination transition rate from band to band, or from bandedge to defect or impurity state, as the case may be, multiplied by the energy difference between initial and final states. ηR is the quantum yield for IR radiative emission upon carrier recombination into one of these states. TT(z,t) is the total temperature, including background temperature, temperature increase due to thermal-wave oscillation following laser-modulated absorption and optical heating, as well as other nonradiative energy conversion pathways. Therefore,WNR[TT(z,t);λ]=WP[Ts(z,t);λ]+(1−ηR)WeR(λ)+WeH(λ)[W/μm]  (2)
Here, WP[Ts(z,t);λ]dλ is the familiar Planck distribution function, or spectral emissive power, representing the rate of radiative recombination within dλ, and sample volume ΔV=A[αIR(λ)]−1 of emitting cross-sectional area A normal to the z-axis in FIG. 2, and depth equal to the optical absorption depth at infrared wavelength λ. αIR(λ) is the IR absorption coefficient at λ and
                                                        W              P                        ⁡                          [                                                                    T                    s                                    ⁡                                      (                                          z                      ,                      t                                        )                                                  ;                λ                            ]                                ⁢          d          ⁢                                          ⁢          λ                =                                            8              ⁢              π              ⁢                                                          ⁢                              h                ⁡                                  (                                                            c                      o                                        /                    n                                    )                                            ⁢              A              ⁢                                                          ⁢              d              ⁢                                                          ⁢              λ                                                      λ                5                            ⁢                              {                                                      exp                    ⁡                                          [                                                                                                    hc                            o                                                    /                          λ                                                ⁢                                                                                                  ⁢                                                  nk                          B                                                ⁢                                                                              T                            s                                                    ⁡                                                      (                                                          z                              ,                              t                                                        )                                                                                              ]                                                        -                  1                                }                                              ⁡                      [            W            ]                                              (        3        )            (co/n) is the speed of light in the medium of refractive index n. Ts(z,t) is made up of only two contributions: background temperature and harmonic optical heating of the lattice at modulation frequency f. The remaining symbols in Eq. (2) have the following meanings: WeH(λ) is the thermal IR photon generation power per unit wavelength due to intraband nonradiative de-excitation of hot carriers with energy hvvis−EG, FIG. 1. (1−ηR) is the nonradiative quantum yield for recombination processes which generate total power WeR(λ) per unit wavelength.
The use of Eq. (3) in describing the thermal emissive power assumes the existence of thermal equilibrium in the semiconductor, a condition known as the Principle of Detailed Balance. It states that the rate of radiative recombination at thermal equilibrium within an emission frequency interval dv, centered at frequency v, is equal to the corresponding generation rate of electron-hole pairs by the thermal radiation field present within the semiconductor [25]. Detailed Balance is, in itself, a statement of Kirchhoff's theorem [24], according to which “for any body in (radiative) thermal equilibrium with its environment, the ratio between the spectral emissive power W(T,λ)dλ and the spectral absorptivity α(T,λ), for a given photon frequency v=c/λ and temperature T, is equal to the spectral emissive power WP(T,λ)dλ, Eq. (3), of the blackbody for the same frequency and temperature”.
Although a semiconductor undergoing harmonic carrier generation is not strictly in thermal equilibrium, it has been shown [19] that in low laser power interactions with electronic carriers, the semiconductor can be considered to be at electronic and thermal equilibrium during the oscillation cycle of the photo-excited carrier-wave as long as i) there exist no intense electromagnetic optical or thermal gradient fields in the semiconductor to upset the quantum configuration of the energy states, driving the structure away from electronic energy equilibrium; ii) upward electronic transitions following optical absorption result in efficient radiative de-excitations with minimal temperature increase of the lattice, or iii) even if significant temperature changes occur due to nonradiative decays which may affect the background temperature of the lattice as in the case of CW generation, however, the temperature oscillation itself amounts to only minimal thermal-wave perturbations with no significant consequence in the structure of the energetic manifold of the semiconductor.
Under these conditions electronic transitions occur essentially adiabatically, with minimum thermal energy exchange interactions across well-defined electronic state densities. It also follows that the higher the oscillation frequency, the greater the adiabatic character of the transition, leading to a stricter validation of Kirchhoff's Law through complete thermal decoupling of the CW oscillator ensemble, as experimentally observed by use of PTR [4]. Therefore, despite the large ambient radiation field oscillations, Eq. (6) is expected to remain essentially valid away from free-carrier density equilibrium in PCR. The absence of cross-coupling in the emitted power of Eq. (1) is a statement of the adiabatic superposition of thermal-infrared (Planck-mediated) emissions through the WNR[TT(z,t);λ] term, and direct electronic infrared emissions through the ηRWeR(λ) term under equilibrium (i.e. constant) baseline temperature and a stationary material energy state manifold characterized by a well-defined Fermi level. A by-product of adiabaticity is that the IR spectra of thermal and recombination emissions are independent of each other, a feature which is central to the realization of PCR.
FIG. 2 shows an elementary slice of thickness dz centered at depth z in a semiconductor slab. The crystal is supported by a backing, but is not necessarily in contact with the backing. A modulated laser beam at angular frequency ω=2πf and wavelength λvis impinges on the front surface of the semiconductor. The super-bandgap radiation is absorbed within a (short) distance from the surface, typically, a few μm, given by [α(λvis)]−1 where α(λvis) is the visible-range absorption coefficient of the pump radiation. The ensuing de-excitation processes generally involve radiative and nonradiative energy release components, resulting in the generation of an IR photon field in the semiconductor involving a relatively broad spectral bandwidth. At thermal and electronic equilibrium, assuming a one-dimensional geometry as a result of a large laser beam spotsize and/or thin sample, the emitted IR photons have equal probability of being directed toward the front or the back surface of the material.
A detailed account of all IR emission, absorption, and reflection processes [19] yields the expression for the total IR emissive power at the fundamental frequency across the front surface of the material in the presence of a backing support which acts both as reflector of semiconductor-generated IR radiation with spectrum centered at λ, and as emitter of backing-generated IR radiation centered at wavelength λb 
                                                                        P                T                            ≈                            ⁢                                                ∫                                      λ                    2                                                        λ                    1                                                  ⁢                                                                  ⁢                                                      ⅆ                                          λ                      ⁡                                              [                                                  1                          -                                                                                    R                              1                                                        ⁡                                                          (                              λ                              )                                                                                                      ]                                                                              ⁢                                      {                                          1                      +                                                                                                    R                            b                                                    ⁡                                                      (                            λ                            )                                                                          [                                                  1                          +                                                                                                                                                                                                                                                                      ⁢                                                                  R                        1                                            ⁡                                              (                        λ                        )                                                              ]                                    )                                ⁢                                                      ɛ                    o                                    ⁡                                      (                    λ                    )                                                  ⁢                                                      ∫                    0                    L                                    ⁢                                      Δ                    ⁢                                                                                  ⁢                                                                  W                        P                                            ⁡                                              (                                                  z                          ,                                                      ω                            ;                            λ                                                                          )                                                              ⁢                                                                                  ⁢                                          ⅆ                      z                                                                                  +                              [                                                                            (                                              1                        +                                                                                                            R                              b                                                        ⁡                                                          (                              λ                              )                                                                                ⁡                                                      [                                                          1                              +                                                                                                R                                  1                                                                ⁡                                                                  (                                  λ                                  )                                                                                                                      ]                                                                                              )                                        ⁢                                                                  W                        O                                            ⁡                                              (                                                                              T                            o                                                    ;                          λ                                                )                                                                              -                                                                                                                                                            ⁢                                                                                    W                        P                                            ⁡                                              (                                                                              T                            b                                                    ,                          λ                                                )                                                              ⁢                                                                  ⅇ                        ⁡                                                  (                                                                                    T                              b                                                        ,                            λ                                                    )                                                                    ⁡                                              [                                                  1                          -                                                                                    R                              1                                                        ⁡                                                          (                              λ                              )                                                                                                      ]                                                                              ]                                ⁢                x                ⁢                                                      ∫                    0                    L                                    ⁢                                                                                    ɛ                        fc                                            ⁢                                                                                          (                                              z                        ,                                                  ω                          ;                          λ                                                                    )                                        ⁢                                          ⅆ                      z                                                                                  }                                                          (        4        )            where R1 is the front surface reflectivity, Rb is the backing support material reflectivity, εo(λ) is the background IR emission coefficient of the material, εfc(z,ω,λ) is the IR emission coefficient due to the free photoexcited carrier wave, e(Tb,λ) is the spectral emissivity of the backing material, ΔWp(z,ω,λ)dz is the harmonic IR emissive power due to the harmonically varying temperature of the sample, Wo(To;λ) is the unmodulated emissive spectral power per unit wavelength due to both Planck-mediated [Wpo(To,λ)] and direct radiative [ηRWeR(λ)] emissions, WP(Tb,λ) is the spectral emissive power per unit wavelength of the backing surface at temperature Tb, and [λ1,λ2] is the spectral bandwidth of the detector. Much work has been done in attempts to separate out carrier-wave and thermal-wave contributions through modulation frequency filtering [2–5], however, they are always strongly mixed and can be separated out effectively only through spectral filtering at the IR detector. The present invention is concerned with the successful separation of the carrier wave from the thermal wave and the instrumental implementation of a technique (“Photo-Carrier Radiometry”) which monitors the former wave in semiconductor materials and devices exclusively.